Matrix Convex Functions With Applications to Weighted Centers for Semidefinite Programming
نویسندگان
چکیده
In this paper, we develop various calculus rules for general smooth matrix-valued functions and for the class of matrix convex (or concave) functions first introduced by Löwner and Kraus in 1930s. Then we use these calculus rules and the matrix convex function − log X to study a new notion of weighted centers for semidefinite programming (SDP) and show that, with this definition, some known properties of weighted centers for linear programming can be extended to SDP. We also show how the calculus rules for matrix convex functions can be used in the implementation of barrier methods for optimization problems involving nonlinear matrix functions.
منابع مشابه
Matrix Functions and Weighted Centers for Semidefinite Programming
In this paper, we develop various differentiation rules for general smooth matrix-valued functions, and for the class of matrix convex (or concave) functions first introduced by Löwner and Kraus in the 1930s. For a matrix monotone function, we present formulas for its derivatives of any order in an integral form. Moreover, for a general smooth primary matrix function, we derive a formula for al...
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